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Stochastic and dynamic vehicle routing with general demand and interarrival time distributions

Part of: Geometric probability and stochastic geometry Special processes

Published online by Cambridge University Press:  01 July 2016

Dimitris J. Bertsimas  and
Garrett Van Ryzin
Dimitris J. Bertsimas*
Affiliation:
Sloan School of Management, MIT
Garrett Van Ryzin*
Affiliation:
Columbia University
*
* Postal address: Sloan School of Management, MIT, Room E53-359, Cambridge, MA 02139, USA.
** Postal address: Graduate School of Business, Columbia University, New York, NY 10027, USA.
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Abstract

We analyze a class of stochastic and dynamic vehicle routing problems in which demands arrive randomly over time and the objective is minimizing waiting time. In our previous work ([6], [7]), we analyzed this problem for the case of uniformly distributed demand locations and Poisson arrivals. In this paper, using quite different techniques, we are able to extend our results to the more realistic case where demand locations have an arbitrary continuous distribution and arrivals follow only a general renewal process. Further, we improve significantly the best known lower bounds for this class of problems and construct policies that are provably within a small constant factor relative to the optimal solution. We show that the leading behavior of the optimal system time has a particularly simple form that offers important structural insight into the behavior of the system. Moreover, by distinguishing two classes of policies our analysis shows an interesting dependence of the system performance on the demand distribution.

Keywords

GENERAL DEMAND DISTRIBUTIONGENERAL INTERARRIVAL TIMESBOUNDSHEURISTICS

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 947 - 978
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

The research of D. Bertsimas was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory. The research of both authors was partially supported by the National Science Foundation under grant DDM-9014751 and by grants from Draper Laboratory and the UPS Foundation.

References

[1] Bartholdi, J. J. and Platzman, L. K. (1982) An O(N log N) planar traveling salesman heuristic based on space filling curves. Operat. Res. Lett. 1, 121–125.Google Scholar
[2] Bartholdi, J. J. and Platzman, L. K. (1988) Heuristics based on space filling curves for combinatorial problems in Euclidean space. Management Sci. 34, 291–305.Google Scholar
[3] Beardwood, J., Halton, J. and Hammersley, J. (1959) The shortest path through many points. Proc. Camb. Phil. Soc. 55, 299–327.Google Scholar
[4] Bertsimas, D. and Nakazato, D. (1991) The general distributional Little's law and its applications. Operat. Res. To appear.Google Scholar
[5] Bertsimas, D. and Van Ryzin, G. (1990) An asymptotic determination of the minimum spanning tree and minimum matching constants in geometrical probability. Oper. Res. Lett. 9, 223–231.CrossRefGoogle Scholar
[6] Bertsimas, D. and Van Ryzin, G. (1990) A stochastic and dynamic vehicle routing problem in the euclidean plane. Operat. Res. 39, 601–615.Google Scholar
[7] Bertsimas, D. and Van Ryzin, G. (1993) Stochastic and dynamic vehicle routing in the euclidean plane with multiple capacitated vehicles. Operat. Res. 41, 60–76.Google Scholar
[8] Haimovich, M. and Rinnooy Kan, A. H. G. (1985) Bounds and heuristics for capacitated routing problems. Math. Operat. Res. 10, 527–542.CrossRefGoogle Scholar
[9] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1934) Inequalities. Cambridge University Press.Google Scholar
[10] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I and II. Adv. App. Prob. 2, 150–177 and 355-364.Google Scholar
[11] Johnson, D. (1988) Talk presented at the Mathematical Programming Symposium, Tokyo.Google Scholar
[12] Karp, R. (1977) Probabilistic analysis of partitioning algorithms for the traveling salesman in the plane. Math. Operat. Res. 2, 209–224.Google Scholar
[13] Karp, R. M. and Steele, J. M. (1985) Probabilistic analysis of heuristics. In The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, ed. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G. and Shmoys, D. B. Wiley, Chichester.Google Scholar
[14] Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G. and Shmoys, D. B., (Eds.) (1985) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester.Google Scholar
[15] Papadimitriou, C. H. (1978) The probabilistic analysis of matching heuristics. Proc. 15th Annual Allerton Conf. on Communication, Control and Computing, pp. 368–378.Google Scholar
[16] Platzman, L. K. and Bartholdi, J. J. (1983) Spacefilling curves and the planar traveling salesman problem. PDRC Technical Report 83-02, Georgia Institute of Technology.Google Scholar
[17] Psaraftis, H. (1988) Dynamic vehicle routing problems. In Vehicle Routing: Methods and Studies, ed. Golden, B. and Assad, A., North-Holland, Amsterdam.Google Scholar
[18] Rhee, W. (1991) On the TSP in many dimensions. Random Structures. To appear.Google Scholar
[19] Steele, J. M. (1981) Subadditive euclidean functionals and nonlinear growth. Ann. Prob. 9, 365–376.Google Scholar
[20] Wolff, R. W. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.CrossRefGoogle Scholar